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Zbl 1073.35178
Messaoudi, Salim A.; Houari, Belkacem Said
Global non-existence of solutions of a class of wave equations with non-linear damping and source terms.
(English)
[J] Math. Methods Appl. Sci. 27, No. 14, 1687-1696 (2004). ISSN 0170-4214; ISSN 1099-1476/e

The authors prove nonexistence of global solutions in time of the initial boundary value problem $$u_{tt}-\Delta u_t- \text{div}(|\nabla u|^{\alpha- 2}\nabla u)- \text{div}(|\nabla u_t|^{\beta- 2}\nabla u_t)+ a|u_t|^{m-2} u_t= b|u|^{p-2} u,\quad x\in\Omega,\ t> 0,$$ $$u(x,0)= u_0(x),\ u_t(x,0)= u_1(x),\ x\in\Omega,\qquad u(x,0)= 0,\ x\in\partial\Omega,\ t> 0,$$ where $a,b> 0$, $\alpha,\beta, m,p> 2$, and $\Omega$ is a bounded domain of $\bbfR^n$ $(n\ge 1)$, with a smooth boundary $\partial\Omega$. This complicated equation is said to appear in nonlinear viscoelasticity theory. The authors suppose that there exist solutions in a function space $Z$, where $$Z= L^\infty([0, T); W^{1,\alpha}_0(\Omega))\cap W^{1,\infty}([0, T); L^2(\Omega)) \cap W^{1,\beta}([0, T); W^{1,\beta}_0(\Omega))\cap W^{1,m}([0, T); L^m(\Omega))$$ for $T> 0$. Local existence and uniqueness are not especially discussed. The energy is defined by $$E(t)= {1\over 2} \int_\Omega u^2_t \,dx+ {1\over\alpha} \int_\Omega |\nabla u|^\alpha \,dx- {b\over p} \int_\Omega |u|^p \,dx.$$ Let $r_\alpha$ denote the Sobolev critical exponent of $W^{1,\alpha}_0(\Omega)$. There is just one theorem. Theorem: Assume that $\beta< \alpha$ and $\max\{m,\alpha\}< p< r_\alpha$. If $E(0)< 0$, then the solution $u\in Z$ cannot exist for all time. The authors insist that their result improves one by {\it Z. Yang} [Math. Methods Appl. Sci. 25, No. 10, 825--833 (2002; Zbl 1009.35051)], whose result requires an additional assumption, $H(0)> A$ (a constant depending on the size of $\Omega$).\par The proof is performed as follows. Define $$L(t)= H^{1-\sigma}(t)+ \varepsilon \int_\Omega uu_t \,dx,$$ where $H(t)= -E(t)$. The main aim of the proof is to show that for suitable $\varepsilon$, $\sigma> 0$ $$L'(t)\ge \xi L^q(t),\quad q> 1.$$ It follows easily from this inequality that $L(t)$ attains $\infty$ in finite time. The mathematical tool of higher grade are not needed, but silful calculation is performed to obtain the inequality. Young's inequality, and Sobolev's inequality, PoincarĂ©s inequaly in bounded domains are exploited.\par Global existence is not especially referred.
[Hiroshi Uesaka (Tokyo)]
MSC 2000:
*35L75 Nonlinear hyperbolic PDE of higher $(>2)$ order
35L35 Higher order hyperbolic equations, boundary value problems

Keywords: nonlinear damping; nonlinear source; negative initial energy; global nonexistence

Citations: Zbl 1009.35051

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